New classification techniques for ordinary differential equations

The goal of the present paper is to propose an enhanced ordinary differential equation solver by exploitation of the powerful equivalence method of Élie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of a dictionary of ODEs, that are regarded as well-known, or at least well-studied. The dictionary considered in this article comprises the ODEs in a book of Kamke. The major advantage of our solver is that the equivalence transformation is obtained without integrating differential equations. We provide also a theoretical contribution revealing the relationship between the change of coordinates that maps two differential equations and their symmetry pseudo-groups.